Semicontinuous Solutions of Hamilton–Jacobi–Bellman Equations with Degenerate State Constraints

Frankowska, Hélène; Plaskacz, Sławomir

doi:10.1006/jmaa.2000.7070

Cited by 47 publications

(64 citation statements)

References 10 publications

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“…In [11,12,15] these results were extended to problems with state constraints. In [11] state constraints given by an arbitrary closed set with nonempty interior were considered and in [12] these results were generalized to degenerate sets. In both papers the multifunction F (t, x) is assumed to be Lipschitz in both t and x and it is assumed that the following outward pointing condition holds, given here, for the sake of simplicity, in the case when Ω has a smooth boundary and nonempty interior:…”

Section: L(t X(t) U(t))dtmentioning

confidence: 99%

“…Characterizations of the Value Function as the unique lower semicontinuous solution of the Hamilton-Jacobi-Bellman equation in the absence of state constraints were provided by Barron and Jensen [2] and Frankowska [9]. In [11,12,15] these results were extended to problems with state constraints. In [11] state constraints given by an arbitrary closed set with nonempty interior were considered and in [12] these results were generalized to degenerate sets.…”

Section: L(t X(t) U(t))dtmentioning

confidence: 99%

“…Lower semicontinuous solutions to Hamilton-Jacobi equations in the presence of state constraints were investigated by Frankowska and Plaskacz (see [11,12]) by using the subdifferential of the value function. However, if the data is only measurable in the time variable, then, like it was done in [13] in the absence of state constraints, one has to adapt the definition of solutions of (HJE) in order to have uniqueness.…”

Section: Denote By V : [0 T ] × ω → R ∪ {+∞} the Value Function Of Tmentioning

confidence: 99%

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Hölder estimates for trajectories of differential inclusions and HJB equations with state constraints

Facchi¹,

Hoehener²

2012

Nonlinear Differ. Equ. Appl.

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Abstract. We consider the differential inclusionẋ ∈ F (t, x), with F measurable in t and Lipschitz in x, together with the state constraint x(t) ∈ Ω and prove the existence of neighboring trajectories lying in the interior of Ω. Using this result, we can characterize the Value Function of the Bolza Problem as the unique lower semicontinuous solution to the corresponding Hamilton-Jacobi-Bellman equation.Mathematics Subject Classification. 34A60, 34H05, 49K15, 49L25, 93C15.

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Section: L(t X(t) U(t))dtmentioning

confidence: 99%

Section: L(t X(t) U(t))dtmentioning

confidence: 99%

Section: Denote By V : [0 T ] × ω → R ∪ {+∞} the Value Function Of Tmentioning

confidence: 99%

See 1 more Smart Citation

Hölder estimates for trajectories of differential inclusions and HJB equations with state constraints

Facchi¹,

Hoehener²

2012

Nonlinear Differ. Equ. Appl.

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“…Recall that several papers have been devoted to study the characterization of the value function for state constrained control problems. Under some controllability assumption and when the set of state-constraints is not time-dependent, the value function can be shown to be the unique constrained-viscosity solution on an adequate HJB equation, see in [28,29,18]. We refer also to [4,1] for a discussion on the general case where the control problem is lacking controllability properties.…”

Section: General Case Without Any Controllability Assumptionmentioning

confidence: 99%

“…Hamilton-Jacobi approach for state-constrained control problems have been extensively studied in the literature [28,29,13,18,4]. When the state constraints are time-dependent, the characterization of the value function becomes more complicated [19].…”

mentioning

confidence: 99%

State-Constrained Optimal Control Problems of Impulsive Differential Equations

Forcadel

Rao²,

Zidani³

2013

Appl Math Optim

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Abstract. The present paper studies an optimal control problem governed by measure driven differential systems and in presence of state constraints. The first result shows that using the graph completion of the measure, the optimal solutions can be obtained by solving a reparametrized control problem of absolutely continuous trajectories but with time-dependent state-constraints. The second result shows that it is possible to characterize the epigraph of the reparametrized value function by a Hamilton-Jacobi equation without assuming any controllability assumption 1 .

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Hamilton–Jacobi–Bellman Equations

Festa

Guglielmi²,

Hermosilla

et al. 2017

Lecture Notes in Mathematics

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Semicontinuous Solutions of Hamilton–Jacobi–Bellman Equations with Degenerate State Constraints

Cited by 47 publications

References 10 publications

Hölder estimates for trajectories of differential inclusions and HJB equations with state constraints

Hölder estimates for trajectories of differential inclusions and HJB equations with state constraints

State-Constrained Optimal Control Problems of Impulsive Differential Equations

Hamilton–Jacobi–Bellman Equations

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